Marché's decomposition conjecture for Kauffman bracket skein modules

Let $M$ be a closed oriented $3$-manifold, and let $\mathcal S_{2,\infty}(M)$ denote its Kauffman bracket skein module over $\mathbb Z[A^{\pm 1}]$. Marché's decomposition conjecture. There exists an integer $d>0$ and finitely generated $\mathbb Z[A^{\pm 1}]$-modules $N_k$ such that

$$\mathcal S_{2,\infty}(M)\cong \mathbb Z[A^{\pm 1}]^d\oplus\bigoplus_{k\geq 1}N_k,$$

where $N_k$ is an $(A^k-A^{-k})$-torsion module for each integer $k$. Marché proposed this as a structural description of skein modules of closed oriented $3$-manifolds, but it has been disproved by counterexamples, including the skein module of the connected sum of two copies of $\mathbb{R}P^3$ and, in this paper, the connected sum of two copies of $S^1\times S^2$.